STRUCTURAL LOGIC AND ABSTRACT ELEMENTARY CLASSES WITH INTERSECTIONS WILL BONEY AND SEBASTIEN VASEY Abstract. We give a syntactic characterization of abstract elementary classes (AECs) closed under intersections using a new logic with a quantifier for iso- morphism types that we call structural logic: we prove that AECs with in- tersections correspond to classes of models of a universal theory in structural logic. This generalizes Tarski's syntactic characterization of universal classes. As a corollary, we obtain that any AEC closed under intersections with count- able L¨owenheim-Skolem number is axiomatizable in L1;!(Q), where Q is the quantifier \there exists uncountably many". Contents 1. Introduction 1 2. Structural quantifiers 3 3. Axiomatizing abstract elementary classes with intersections 7 4. Equivalence vs. Axiomatization 11 References 13 1. Introduction 1.1. Background and motivation. Shelah's abstract elementary classes (AECs) [She87,Bal09,She09a,She09b] are a semantic framework to study the model theory of classes that are not necessarily axiomatized by an L!;!-theory. Roughly speak- ing (see Definition 2.4), an AEC is a pair (K; ≤K) satisfying some of the category- theoretic properties of (Mod(T ); ), for T an L!;!-theory. This encompasses classes of models of an L1;! sentence (i.e. infinite conjunctions and disjunctions are al- lowed), and even L1;!(hQλi ii<α) theories, where Qλi is the quantifier \there exists λi-many". Since the axioms of AECs do not contain any axiomatizability requirement, it is not clear whether there is a natural logic whose class of models are exactly AECs. More precisely, one can ask whether there is a natural abstract logic (in the sense of Barwise, see the survey [BFB85]) so that classes of models of theories in that logic are AECs and any AEC is axiomatized by a theory in that logic. Date: December 9, 2018 AMS 2010 Subject Classification: Primary 03C48. Secondary: 03B60, 03C80, 03C95. Key words and phrases. Abstract elementary classes; Intersections; Universal classes; Struc- tural logic. This material is based upon work done while the first author was supported by the National Science Foundation under Grant No. DMS-1402191. 1 2 WILL BONEY AND SEBASTIEN VASEY An example of the kind of theorem one may expect is Tarski's characterization of universal classes. Tarski showed [Tar54] that classes of structures in a finite relational vocabulary which are closed under isomorphism, substructures, and union of chains (according to the substructure relation) are exactly the classes of models of a universal L!;! theory. The proof of Tarski's result generalizes to non-finite vocabularies as follows: Definition 1.1. K is a universal class if it is a class of structures in a fixed vocabulary that is closed under isomorphisms, substructures, and unions of chains (according to the substructure relation). Fact 1.2 (Tarski's presentation theorem, [Tar54]). Let K be a class of structures in a fixed vocabulary. The following are equivalent: (1) K is a universal class. (2) K is the class of models of a universal L1;!theory. Here, a universal sentence is one of the form 8x0 : : : xn−1 with 2 L1;! quantifier-free. Note that this is not the only definition of universal sentences in the literature; see [Vas17b, Remark 2.5] for further discussion. Universal classes are a special type of abstract elementary classes. In a sense, their complexity is quite low and indeed several powerful theorems can be proven there (see e.g. [She09b, Chapter V] and the second author's ZFC proof of the eventual categoricity conjecture there [Vas17a,Vas17b]). A more general kind of AECs are AECs with intersections. They were intro- duced by Baldwin and Shelah [BS08, Definition 1.2]. They are defined as the AECs in which the intersection of any set of K-substructures of a fixed model N is again a K-substructure, see Definition 3.1. In universal classes, this property follows from closure under substructure so any universal class is an AEC with intersec- tions. Being closed under intersections does not imply that the classes are easy to analyze, e.g. [HS90,BK09,BS08,BU17] provide examples of AECs closed under in- tersections that fail to be tame. Nevertheless, AECs with intersections are still less complex than general AECs. For example, the second author has shown that She- lah's eventual categoricity conjecture holds there assuming a large cardinal axiom [Vas17a, Theorem 1.7], whereas the conjecture is still open for general AECs. 1.2. Tarski's presentation theorem for AECs with intersections. In the present paper, we generalize Tarski's presentation theorem to AECs with inter- κ-struct sections as follows: we introduce a new logic, L1;! , which is essentially L1;! expanded by what we call structural quantifiers, and show that AECs with inter- sections are (essentially) exactly the class of models of a particular kind of theory{ struct κ-struct what we call a 8Q -theory (Definition 2.8){in the logic L1;! . More precisely struct κ-struct (Corollary 3.11), any 8Q -theory in L1;! gives rise to an AEC with intersec- tions and, for any AEC with intersections, there is an expansion of its vocabulary with countably-many relation symbols so that the resulting class is axiomatized by struct κ-struct a 8Q -theory in L1;! . Moreover the expansion is functorial (i.e. it induces an isomorphism of concrete category, see Definition 3.3). The idea of the proof is to code the isomorphism types of the set c`N (a), where c`N (a) denotes the intersections of all the K-substructures of N containing the finite κ-struct sequences a. The logic L1;! is expanded by a family of generalized quantifiers in the sense of Mostowski and Lindstr¨om[Mos57,Lin66]. We also add quantifiers STRUCTURAL LOGIC AND ABSTRACT ELEMENTARY CLASSES WITH INTERSECTIONS 3 such as Qstruct xyφ(x) (y), which asks whether the solution sets (A; B) of (φ, ) (M2;M1) are isomorphic to (M2;M1) (where of course the two isomorphisms must agree). This is crucial to code the ordering of the AEC. Our characterization also gener- alizes Kirby's result on the definability of Zilber's quasiminimal classes [Kir10, x5]. Indeed it is easy to see (Remark 2.3) that @1-struct is just (Q) (where Q is the L!1;! L!1;! quantifier \there exists uncountably many") and quasiminimal classes are in par- ticular AECs with intersections, see [Vas18]. Thus we obtain that any AEC with intersections and countable L¨owenheim-Skolem-Tarski number is axiomatizable in L1;!(Q), see Corollary 3.12. An immediate conclusion is that any AEC which admits intersections, has L¨owenheim- Skolem-Tarski number @0, and has countably-many countable models, has a Borel functorial expansion (in the sense that its restriction to @0 can be coded by a Borel set of reals, see Corollary 3.14). This is further evidence for the assertion that AECs with intersections have low complexity and paves the way for the use of de- scriptive set-theoretic tools in the study of these classes (see [She09a, Chapter I], [BLS15,BL16]). 1.3. Other approaches. Rabin [Rab62] syntactically characterizes L!;! theories whose class of models is closed under intersections. The characterization is (prov- ably) much more complicated than that of universal classes. This paper shows that by changing the logic we can achieve a much easier characterization. In a recent preprint [LRV], Lieberman, Rosick´y,and the second author have shown that any AEC with intersections is a locally @0-polypresentable category. In particular, it is @0-accessible, and this implies that it is equivalent (as a category) to a class of models of an L1;! sentence. We discuss these approaches in greater detail in Section 4. 1.4. Notation. We denote the universe of a τ-structure M by jMj and its cardi- − nality by kMk. We use κ to denote the predecessor of a cardinal κ: it is κ0 if + κ = κ0 and κ otherwise. 1.5. Acknowledgments. We thank John T. Baldwin, Marcos Mazari-Armida, and the referee for feedback which helped improve the presentation of this paper. 2. Structural quantifiers κ-struct We define a new logic, L . It consists of L!;! with a family of quantifiers struct I(M) QM;A , which generalize the quantifier Q from [MSS76, x4]. Compared to I(M) Q , we allow multiple formulas and τ0-structures, with τ0 a sub-vocabulary of τ. Definition 2.1. Let τ be a vocabulary and κ be an infinite cardinal. We define the logic Lκ-struct(τ) as follows: (1) Lκ-struct is the smallest set closed under the following: (a) Atomic formulas; (b) Negation; (c) Binary conjunction and disjunction; (d) Existential and universal quantification; and κ-struct (e) If n < !, φ(x; z) and h i(yi; z): i < ni are formulas in L (τ), τ0 is a sub-vocabulary of τ, M is a τ0-structure with universe an ordinal 4 WILL BONEY AND SEBASTIEN VASEY strictly less than κ and A := hAi : i < ni are subsets of jMj, then letting y := hyi : i < ni: struct κ-struct QM;A xyφ(x; z)h i(yi; z): i < ni 2 L (τ) κ-struct (2) Satisfaction L is defined inductively as follows (we omit the subscript since it will always be clear from context): (a) As usual for the first-order operations. (b) If N is a τ-structure, n < !, φ(x; z) and h i(yi; z): i < ni are formu- κ-struct las in L (τ), τ0 is a sub-vocabulary of τ, M is a τ0-structure with universe an ordinal strictly less than κ and A := hAi : i < ni are sub- struct sets of jMj, then letting y := hyi : i < ni, N j= QM;A xyφ(x; b)h i(yi; b): i < ni if and only if: (i) For all i < n, N j= 8x ( i(x; b) ! φ(x; b)).